Use the given sum or difference identity to find cos⁡(α−β)\displaystyle {\cos{{\left(\alpha-\beta\right)}}} exactly, given that sin⁡α=−17\displaystyle {\sin{\alpha}}=-\frac{{1}}{{7}} such that α\displaystyle \alpha terminates in quadrant III and sin⁡β=47\displaystyle {\sin{\beta}}=\frac{{4}}{{7}} such that β\displaystyle \beta terminates in quadrant II.

  cos⁡(α−β)=cos⁡(α)cos⁡(β)+sin⁡(α)sin⁡(β)\displaystyle {\cos{{\left(\alpha-\beta\right)}}}={\cos{{\left(\alpha\right)}}}{\cos{{\left(\beta\right)}}}+{\sin{{\left(\alpha\right)}}}{\sin{{\left(\beta\right)}}} 

 cos⁡(α−β)=\displaystyle {\cos{{\left(\alpha-\beta\right)}}}= (\displaystyle {(} )(\displaystyle {)}{(} )\displaystyle {)} +\displaystyle + (−17)(47)\displaystyle {\left(-\frac{{1}}{{7}}\right)}{\left(\frac{{4}}{{7}}\right)} 

 cos⁡(α−β)=\displaystyle {\cos{{\left(\alpha-\beta\right)}}}=

 

Your answers should be exact but you do not have to rationalize your denominators.

License